非互易系统的广义布里渊区
Generalised Brillouin Zone for Non-Reciprocal Systems
AI总结：
本文提出复准周期性的广义布里渊区，严格解释非互易系统本征模的单向衰减与谱收敛，并推广 Hermitian 理论及 $k$-Toeplitz 矩阵情形。
摘要：
近来，人们观察到具有实准周期性的 Floquet-Bloch 变换无法刻画非互易系统的谱性质。本文旨在通过允许准周期性取复数来引入广义布里渊区的概念，以弥补这一缺陷。文中证明，将布里渊区移入复平面可以解释本征模的单向空间衰减，并带来正确的谱收敛性质。本文的结果阐明并严格证明了有限结构的谱性质如何与相应的半无限或无限周期晶格的谱性质相关联，并明确刻画了如何将 Hermitian 理论推广到非互易情形。基于我们的理论，我们分别刻画了开放边界条件和周期边界条件下的广义布里渊区。我们的结果与物理文献一致，并给出了对 $k$-Toeplitz 矩阵情形的明确推广。
英文摘要：
Recently, it has been observed that the Floquet-Bloch transform with real quasiperiodicities fails to capture the spectral properties of non-reciprocal systems. The aim of this paper is to introduce the notion of a generalised Brillouin zone by allowing the quasiperiodicities to be complex in order to rectify this. It is proved that this shift of the Brillouin zone into the complex plane accounts for the unidirectional spatial decay of the eigenmodes and leads to correct spectral convergence properties. The results in this paper clarify and prove rigorously how the spectral properties of a finite structure are associated with those of the corresponding semi-infinitely or infinitely periodic lattices and give explicit characterisations of how to extend the Hermitian theory to non-reciprocal settings. Based on our theory, we characterise the generalised Brillouin zone for both open boundary conditions and periodic boundary conditions. Our results are consistent with the physical literature and give explicit generalisations to the $k$-Toeplitz matrix cases.
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